三角函数
和差化积公式
$$
\begin{aligned}
&[1]. \sin(\alpha+\beta) = \sin \alpha \cos\beta+\cos \alpha\sin \beta &&[2]. \sin(\alpha-\beta) = \sin \alpha\cos \beta-\cos \alpha \sin \beta\\
&[3]. \cos(\alpha+\beta) = \cos \alpha\cos\beta-\sin \alpha\sin \beta &&[4].
\cos(\alpha-\beta) = \cos \alpha\cos\beta+\sin \alpha\sin \beta\\
&[5]. \tan(\alpha+\beta) = \frac{\tan \alpha \tan \beta}{1-\tan \alpha\tan\beta}\\
\end{aligned}
$$
积化和差公式
$$
\begin{aligned}
&[1]. \sin \alpha \cos\beta = \frac12[\cos(\alpha-\beta)-\cos(\alpha+\beta)] && [2]. \sin \alpha \cos \beta = \frac12 [\sin (\alpha-\beta)+\sin(\alpha+\beta)]\\
&[3]. \cos \alpha \sin \beta = \frac12 [\cos(\alpha-\beta)+\cos(\alpha+\beta)]
\end{aligned}
$$
倍角公式
$$
\begin{aligned}
&[1]. \sin 2\alpha = 2\sin \alpha \cos \alpha &&[2]. \cos 2\alpha=\cos^2\alpha-\sin^2\alpha=1-2\sin^2\alpha=2\cos^2\alpha-1\\
&[3]. \tan 2\alpha = \frac{2\tan \alpha}{1-\tan^2\alpha}
\end{aligned}
$$
极限
两个重要极限
$$
\begin{aligned}
&[1]. \lim_{x\to 0}\frac{\sin x}{x} =1 &&[2]. \lim_{x\to \infty}(1+\frac 1x)^x = \lim_{t\to 0}(1+t)^\frac1t = e\\
\end{aligned}
$$
常用等价无穷小
$$
\begin{aligned}
&[1]. \quad \sin x\sim x \quad (x\to 0) &&[2]. \quad \tan x \sim x \quad (x\to 0)
\\
&[3]. \quad \arcsin x \sim x \quad (x\to 0) &&[4]. \quad 1-\cos x \sim \frac12 x^2 \quad (x\to 0)\\
&[5]. \quad (1+x)^\alpha \sim \alpha x \quad (x\to 0) \quad \alpha \in R &&[6]. \quad e^x -1 \sim x \quad (x\to 0)\\
&[7]. \quad \ln (1+x)\sim x \quad (x\to 0) \\
\end{aligned}
$$
导数
$$
\begin{aligned}
&[1]. \quad C^\prime = 0 \quad C\in R && [2]. \quad (x^\mu)^\prime = \mu x^{\mu-1}\\
&[3]. \quad (a^x)^\prime = a^x \ln a(a>0且 a\neq 1) &&[4]. \quad(e^x)^\prime = e^x\\
&[5]. \quad(\log_a x )^\prime = \frac{1}{x \ln a} (a>0且 a\neq 1) &&[6]. \quad(\ln x)^\prime = \frac1x\\
&[7]. \quad(\sin x)^\prime = \cos x && [8]. \quad(\cos x)^\prime = - \sin x\\
&[9]. \quad(\tan x)^\prime = \sec^2 x && [10]. \quad(\sec x)^\prime = \sec x\tan x \\
&[11]. \quad(\cot x)^\prime = - \csc^2 x &&[12]. \quad(\csc x)^\prime = -\csc x\cot x\\
&[13]. \quad(\arcsin x)^\prime = \frac{1}{\sqrt{1-x^2}} &&[14]. \quad(\arccos x)^\prime = -\frac{1}{\sqrt{1-x^2}} \\
&[15]. \quad(\arctan x)^\prime = \frac{1}{1+x^2} &&[16]. \quad(arccot \quad x)^\prime = -\frac{1}{1+x^2}
\end{aligned}
$$
常用泰勒展开
$$
\begin{aligned}
&[1].\frac{1}{1-x} = 1+x+x^2+x^3+ \cdots = \sum^{\infty}_{n=0}x^n &&x \in (-1,1)
\end{aligned}
$$
$$
\begin{aligned}
&[2].\frac{1}{1+x}=1-x+x^2-x^3+\cdots = \sum^{\infty}_{n=0}(-1)^nx^n &&x \in (-1,1)
\end{aligned}
$$
$$
\begin{aligned}
&[3]. e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots = \sum_{n=0}^{\infty}\frac{x^n}{n!} &&x\in (-\infty,+\infty)
\end{aligned}
$$
$$
\begin{aligned}
&[4]. \sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+ \cdots = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!} && x\in (-\infty,+\infty)
\end{aligned}
$$
$$
\begin{aligned}
&[5]. \cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots = \sum_{n=0}^{\infty}(-1)^n \frac{x^{2n}}{(2n)!} &&x\in (-\infty,+\infty)
\end{aligned}
$$
$$
\begin{aligned}
&[6]. \tan^{-1}x = x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots = \sum_{n=0}^{\infty}(-1)^n \frac{x^{2n+1}}{2n+1} && x\in [-1,1]
\end{aligned}
$$
$$
\begin{aligned}
&[7]. \ln(x+1) = x-\frac{x^2}{2}+\frac{x^3}{3}+\cdots = \sum_{n=1}^{\infty}(-1)^n\frac{x^{n+1}}{n+1} && x\in (-1,1]
\end{aligned}
$$
$$
\begin{aligned}
&[8]. (1+x)^\alpha = 1+\alpha x + \frac{\alpha(\alpha-1)}{2!}x^2+\frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3 + \cdots = \sum_{n=0}^{\infty} C_n^\alpha x^n && x \in (-1,1),\alpha \in R
\end{aligned}
$$
基本积分表
$$
\begin{aligned}
&[1].\quad \int k dx = kx+C &&[2]. \quad \int x^\mu dx = \frac{x^{\mu+1}}{\mu+1}+C (\mu\neq -1)\\
&[3]. \quad \int \frac{dx}{x}dx = \ln |x|+C \qquad \qquad &&[4]. \quad \int \frac{dx}{1+x^2} dx = \arctan x+C\\
&[5]. \quad \int \frac{dx}{\sqrt{1-x^2}} dx = \arcsin x+C &&[6]. \quad \int \cos dx = \sin x+C\\
&[7]. \quad \int \sin x dx = -\cos x+C &&[8]. \quad\int \sec^2x dx = \tan x+C \\
&[9]. \quad \int \csc^2 xdx = -\cot x +C &&[10]. \quad \int \sec x\tan x dx = \sec x+C\\
&[11]. \quad \int \csc x \cot x dx = -\csc x+C &&[12]. \quad\int e^x dx = e^x +C\\
&[13]. \quad \int a^x dx = \frac{a^x}{\ln a} +C &&[14]. \int \tan x dx = -\ln |\cos x|+C\\
&[15]. \int \cot xdx = \ln |\sin x|+C &&[16]. \int \sec x dx = \ln|\sec x+ \tan x|+C\\
&[17]. \int \csc xdx = \ln |\csc x- \cot x|+C &&[18]. \int \frac{dx}{a^2+x^2}= \frac1a \arctan \frac xa +C \\
&[19]. \int \frac{dx}{x^2-a^2} = \frac{1}{2a} \ln|\frac{x-a}{x+a}| +C &&[20]. \int \frac{dx}{\sqrt{a^2-x^2}}= \arcsin \frac xa +C\\
&[21]. \int \frac{dx}{\sqrt{a^2+x^2}} = \ln (x+\sqrt{x^2+a^2})+C &&[22]. \int \frac{dx}{\sqrt{x^2-a^2}} = \ln|x+\sqrt{x^2-a^2}| +C\\
\end{aligned}
$$